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Diffusion charging of nonspherical particles by bipolar ions under external magnetic fields
J. Aerosol Sci., Vol. 18. No. 6, pp. 765-768, 1987. Printed in Great Britain
0021--8502/87$3.00+0.00 Pergamon JournalsLtd.
DIFFUSION CHARGING OF NONSPHERICAL PARTICLES BY BIPOLAR IONS UNDER EXTERNAl, MAGNETIC FIELDS
J.S.Chang and S Ono* Department ofEngineering Physics McMaster University Hamilton, Ontario,Canada LSS 4MI *On leave from the Department of Electrical and Electronics Engineering, Musashi Institute of Technology, Tokyo, Japan Introduction Experimental and computer simulations have been conducted for the diffusion charging of the nonspherical particles by bipolar ions under external magnetic fields. These conditions have increased in importance as result of recent developments in high temperature superconductive materials and the high magnetic field technology (Chang 1984). Theory The charged particle transport equations near the aerosol particle are governed by the continuum flux equation for the present effective Knudsen number (Kn < 10-1) condition as follows (Lassen, 1961; Laframbeise and Chang, 1977): J = ::1::p N VV-V • (D N);
V-J=0
[1]
[21 V2V = 0 where J is the ion number flux, N is the ion number density, V is the electric field potential 15.56)<10 -6 Np/Rp; Rp in [cm], Np is the surface charge density, and D and p are the diffusion coefficient and the mobility, respectively The boundary conditions presently considered are: N = 0,
V = Vpat the surface
N --* 0, V -* 0 at infinitedistance For the Maxwell molecules with constant collision frequency Vc, and cyclotron frequency WB = eBtmv, Allis (1956) obtained,
D=
kT rove
-
1
fl
l+fl 2
l+f~ 2
f~ l+f/2
1 l+f/2
0 [31
V
0
0 0 1 with the Einstein relation, D~j = pijkT/e. Here k is the Boltzmann constant, m is the ion mass, B is the external magnetic field strength, T is the temperature, e is the electric charge, fl is the w~vc, v is the thermal velocity (SkT/nm) 1/2. The charged particle transport equations fc, ions m a constant magnetic field aligned with the z axis are, in dimensionless form,
where n = N/N®,
d~ = eV/kT,
x --
X/Rp,
y ~ Y/Rp,
z = Z/Rp,
t = L/2Rp.
765
766
J.S. CHArqo and S. ONo
We consider ellipsoids of revolution of ratio of semi-axes e inclined at angle 0 with respect to the magnetic field (see Figure 1). The axis of revolution of the particle and the magnetic axis (z) define the x-z plane. Thus, the particle P is given by
1
P: (x cos0-(1 +f~2~l/2zsin0)2 +y2 + ~ [ x sin0 +(l+f~2)l/2zcosO] 2 = I
[51
IE 9O
B) ~ a ~=1
i
30
0=1 ~=5
13
ep=O
0.$
1
I .... I 0.1
Fig. 1
1
10
I
I,
t
II
II
0
The effect of external magnetic field on the deposition current at zero surface potentials (or zero surface charge).
where the effect of magnetic field in equation [4] is absorbed in z-coordinate by z = z/(1 + fl2)1/2 (Cohen, 1969). For the limit of dpp = 0, typical numerical results for various t , ~ and 0 are shown in Figure 1, where nonmagnetic field deposition current for the ellipsoidal aerosol particles is (Laframboise and Chang, 1977):
Io = D N® e Rp G(~) G(e) = 4n(1 - e2)l/2/arc cos e
for(e ~ 1)
[6]
G(e) = 4 n ( ~ - 1 ) ~ / ~ n ~ [ ( e + 1) u2 + ( e - 1)1/2]/[(e+ 1 ) ~ - ( e - 1)1~]} for(e ~ 1) Figure 1 shows that the deposition flux decreases with increasing f~, and the minimum flux obtained in the 0 = 0 conditions. No significant e effects on the 0 dependence when e =" 100, and the magnetic field effect can be neglected when 0 = 90 ° under these conditions. Equations [2], [4] and [5] also show that the surface potential or charging conditions independent of magnetic field in the Laplace limit under continuum conditions, i e i = (I(fl)/Io) = di~p/[exp(Cp- 1)] (Laframboise and Chang, 1977). Experimental Simulations The experimental simulation was conducted in two parts. First, conductive d u m m y particle was suspended in a thin shielded wire and the charged particle current flux was mesured under the reduced gas pressure bipolar ionic environments, as shown in Figure 2a, i.e. ions and electrons with Ni® = Ne®. The effect of particle surface charges was simulated by imposing an electric potential on the dummy particles. Then the suspending wire was surrounded by an external magnetic field generated by coils. The particle charging characteristics are obtained for the effective Knudsen number Kn from 1 0 , 4 t o 10 -1, the effective external magnetic field factor fl from 0 to 102 and the shape factore ~ from 0.4 to 40. Typical geometry and size considered here are shown in Figure 2b, which corresponds to Knudsen number for micron particles in atmospheric conditions, i.e. the simulation gas pressure of Argon (0.5 < P ~; 4 Torr) for electrons.
Diffusion charging of nonsphcrical particles
767
A
B
C
-. f.._i,..-
.75;
1
.i
63 mm
)
.75 .1
Fig. 2
Schematics of experimental simulation system and dummy particle dimension=. D: dummy particles; M: electromagnetic coils; F: Faraday cups; A: cylindrical dummy; B: Spherical dummy; C: Bispherical dummy particles.
Experimentally obtained deposition flux as a function of surface potential for various external magnetic fields and shapes are shown in Figures 3a and 3b for attracting and retarding potentials, respectively. Figure 3 shows that no significant magnetic field and dummy particle shape effects on the surface potential dependence of the deposition flux as has been expected from the theory.
I I I I I I J I I 1
I AI I I Ill]
,0 _
_
ii = o s
l,J
...o**,
"
I
I I I IEFf'l
..,/I
']
.~.,/~/I
k
o'o A '1 "'B~ qS[Tl-
__~\
..R
--
i
.
ott]
_
~ ~"
_
101- ~\ ~
i
=
".~
~ \
-
_-
.
I I till]
10"11 0
I 2
I
I~1 4.d)
I\1 6
I 8
I
"Jr
Fig. 3
Deposition current flux as a function of surface potential for various external magnetic fields: a) Attracting; b) Retarding potentials
Concluding R~marks Experimental and theoretical studies have been conducted for the diffusion charging of nonspherical particles by bipolar ions under external magnetic field. The results show that: (1) the effect of magnetic field charging can be neglected when the effective electric field factor 0.1; (2)the effect of particle shape can be estimated by the surface area for the shape factor up to 40 when the particle shape can be approximated by spheroidal shapes; (3) the effect of angles between magnetic field and spheroid axis significantly influenced the particle charging characteristics and the effect increases with decreasing shape factors; (4) the external magnetic field factor= observed to be not significantly influenced by surface potential dependent ofdeposition flux. Acknowledgements This work is suportedpartlyby the Natural Sciencesand Engineering Research Council of Canada and Environmental Research Co.
768
J.S. CHANGand S. ONo
References Allis, W.P. (1956) Handbuch der Physik (Edited by S. Fl~gge ), Springer-Verlag, Berlin, 21. p. 394. Cohen, I.M. (1969) Phys. Fluids, 12, 2356-2361. Chang, J.S. (1984) "Electrical and Magnetic Separation and Filtration Technology" (Edited by R. Vanbrabant and R. Wieler), Koninklijke Vlaamse Ing. Press, pp. 37-42. Laframboise, J.G.and Chang, J.S. (1977), d. Aerosol Sci., 8, 331-338.
0021--8502/87$3.00+0.00 Pergamon JournalsLtd.
DIFFUSION CHARGING OF NONSPHERICAL PARTICLES BY BIPOLAR IONS UNDER EXTERNAl, MAGNETIC FIELDS
J.S.Chang and S Ono* Department ofEngineering Physics McMaster University Hamilton, Ontario,Canada LSS 4MI *On leave from the Department of Electrical and Electronics Engineering, Musashi Institute of Technology, Tokyo, Japan Introduction Experimental and computer simulations have been conducted for the diffusion charging of the nonspherical particles by bipolar ions under external magnetic fields. These conditions have increased in importance as result of recent developments in high temperature superconductive materials and the high magnetic field technology (Chang 1984). Theory The charged particle transport equations near the aerosol particle are governed by the continuum flux equation for the present effective Knudsen number (Kn < 10-1) condition as follows (Lassen, 1961; Laframbeise and Chang, 1977): J = ::1::p N VV-V • (D N);
V-J=0
[1]
[21 V2V = 0 where J is the ion number flux, N is the ion number density, V is the electric field potential 15.56)<10 -6 Np/Rp; Rp in [cm], Np is the surface charge density, and D and p are the diffusion coefficient and the mobility, respectively The boundary conditions presently considered are: N = 0,
V = Vpat the surface
N --* 0, V -* 0 at infinitedistance For the Maxwell molecules with constant collision frequency Vc, and cyclotron frequency WB = eBtmv, Allis (1956) obtained,
D=
kT rove
-
1
fl
l+fl 2
l+f~ 2
f~ l+f/2
1 l+f/2
0 [31
V
0
0 0 1 with the Einstein relation, D~j = pijkT/e. Here k is the Boltzmann constant, m is the ion mass, B is the external magnetic field strength, T is the temperature, e is the electric charge, fl is the w~vc, v is the thermal velocity (SkT/nm) 1/2. The charged particle transport equations fc, ions m a constant magnetic field aligned with the z axis are, in dimensionless form,
where n = N/N®,
d~ = eV/kT,
x --
X/Rp,
y ~ Y/Rp,
z = Z/Rp,
t = L/2Rp.
765
766
J.S. CHArqo and S. ONo
We consider ellipsoids of revolution of ratio of semi-axes e inclined at angle 0 with respect to the magnetic field (see Figure 1). The axis of revolution of the particle and the magnetic axis (z) define the x-z plane. Thus, the particle P is given by
1
P: (x cos0-(1 +f~2~l/2zsin0)2 +y2 + ~ [ x sin0 +(l+f~2)l/2zcosO] 2 = I
[51
IE 9O
B) ~ a ~=1
i
30
0=1 ~=5
13
ep=O
0.$
1
I .... I 0.1
Fig. 1
1
10
I
I,
t
II
II
0
The effect of external magnetic field on the deposition current at zero surface potentials (or zero surface charge).
where the effect of magnetic field in equation [4] is absorbed in z-coordinate by z = z/(1 + fl2)1/2 (Cohen, 1969). For the limit of dpp = 0, typical numerical results for various t , ~ and 0 are shown in Figure 1, where nonmagnetic field deposition current for the ellipsoidal aerosol particles is (Laframboise and Chang, 1977):
Io = D N® e Rp G(~) G(e) = 4n(1 - e2)l/2/arc cos e
for(e ~ 1)
[6]
G(e) = 4 n ( ~ - 1 ) ~ / ~ n ~ [ ( e + 1) u2 + ( e - 1)1/2]/[(e+ 1 ) ~ - ( e - 1)1~]} for(e ~ 1) Figure 1 shows that the deposition flux decreases with increasing f~, and the minimum flux obtained in the 0 = 0 conditions. No significant e effects on the 0 dependence when e =" 100, and the magnetic field effect can be neglected when 0 = 90 ° under these conditions. Equations [2], [4] and [5] also show that the surface potential or charging conditions independent of magnetic field in the Laplace limit under continuum conditions, i e i = (I(fl)/Io) = di~p/[exp(Cp- 1)] (Laframboise and Chang, 1977). Experimental Simulations The experimental simulation was conducted in two parts. First, conductive d u m m y particle was suspended in a thin shielded wire and the charged particle current flux was mesured under the reduced gas pressure bipolar ionic environments, as shown in Figure 2a, i.e. ions and electrons with Ni® = Ne®. The effect of particle surface charges was simulated by imposing an electric potential on the dummy particles. Then the suspending wire was surrounded by an external magnetic field generated by coils. The particle charging characteristics are obtained for the effective Knudsen number Kn from 1 0 , 4 t o 10 -1, the effective external magnetic field factor fl from 0 to 102 and the shape factore ~ from 0.4 to 40. Typical geometry and size considered here are shown in Figure 2b, which corresponds to Knudsen number for micron particles in atmospheric conditions, i.e. the simulation gas pressure of Argon (0.5 < P ~; 4 Torr) for electrons.
Diffusion charging of nonsphcrical particles
767
A
B
C
-. f.._i,..-
.75;
1
.i
63 mm
)
.75 .1
Fig. 2
Schematics of experimental simulation system and dummy particle dimension=. D: dummy particles; M: electromagnetic coils; F: Faraday cups; A: cylindrical dummy; B: Spherical dummy; C: Bispherical dummy particles.
Experimentally obtained deposition flux as a function of surface potential for various external magnetic fields and shapes are shown in Figures 3a and 3b for attracting and retarding potentials, respectively. Figure 3 shows that no significant magnetic field and dummy particle shape effects on the surface potential dependence of the deposition flux as has been expected from the theory.
I I I I I I J I I 1
I AI I I Ill]
,0 _
_
ii = o s
l,J
...o**,
"
I
I I I IEFf'l
..,/I
']
.~.,/~/I
k
o'o A '1 "'B~ qS[Tl-
__~\
..R
--
i
.
ott]
_
~ ~"
_
101- ~\ ~
i
=
".~
~ \
-
_-
.
I I till]
10"11 0
I 2
I
I~1 4.d)
I\1 6
I 8
I
"Jr
Fig. 3
Deposition current flux as a function of surface potential for various external magnetic fields: a) Attracting; b) Retarding potentials
Concluding R~marks Experimental and theoretical studies have been conducted for the diffusion charging of nonspherical particles by bipolar ions under external magnetic field. The results show that: (1) the effect of magnetic field charging can be neglected when the effective electric field factor 0.1; (2)the effect of particle shape can be estimated by the surface area for the shape factor up to 40 when the particle shape can be approximated by spheroidal shapes; (3) the effect of angles between magnetic field and spheroid axis significantly influenced the particle charging characteristics and the effect increases with decreasing shape factors; (4) the external magnetic field factor= observed to be not significantly influenced by surface potential dependent ofdeposition flux. Acknowledgements This work is suportedpartlyby the Natural Sciencesand Engineering Research Council of Canada and Environmental Research Co.
768
J.S. CHANGand S. ONo
References Allis, W.P. (1956) Handbuch der Physik (Edited by S. Fl~gge ), Springer-Verlag, Berlin, 21. p. 394. Cohen, I.M. (1969) Phys. Fluids, 12, 2356-2361. Chang, J.S. (1984) "Electrical and Magnetic Separation and Filtration Technology" (Edited by R. Vanbrabant and R. Wieler), Koninklijke Vlaamse Ing. Press, pp. 37-42. Laframboise, J.G.and Chang, J.S. (1977), d. Aerosol Sci., 8, 331-338.